Seamless animation of Wave procedural texture
You wrong. Perfect number is 1,884955781 ( or Pi / 5 * value in modulo )

For answer the question, first you need to know, that exactly Wave Texture node do.

So I build in simpler nodes exact copy of Wave Texture node:



This node setup produce similar result as Wave Texture node, and you can check this:



So now we can follow the nodes, to know where 1.8888 is hidden.

We begin in modulo coordinates here, from reference question. As i see, you have 3 repeats, so I think you set 0.3333 in modulo. You didn't give any information about it, but it's looks like that I'm right.

 

x,y,z now is from 0 to 0.3333.

Next stop is my node setup, in first nodes texture coordinates are multiplying with the result of this two:



First, multiply 10: 0.3333*10=3.3333

Next it is our X-number, that you think is 1.8888. I call him X, so result is 3.3333*X.



Last stop - sine. You need to know, that Sine repeats in 2*Pi period. So now we can get eaquation:

3.3333*X=2*Pi


So X will be:

X=(2*Pi)/3.3333

X=1,884955781


So, you wrong. perfect number is 1,884955781